Question: Given $ \overrightarrow{OA}\perp\overrightarrow{OC}$, $ m \angle AOB = 7x - 46$, and $ m \angle BOC = 5x - 44$, find $m\angle AOB$. $O$ $A$ $C$ $B$
Answer: From the diagram, we see that together ${\angle AOB}$ and ${\angle BOC}$ form ${\angle AOC}$ , so $ {m\angle AOB} + {m\angle BOC} = {m\angle AOC}$ Since we are given that $\overrightarrow{OA}\perp\overrightarrow{OC}$ , we know ${m\angle AOC = 90}$ Substitute in the expressions that were given for each measure: $ {7x - 46} + {5x - 44} = {90}$ Combine like terms: $ 12x - 90 = 90$ Add $90$ to both sides: $ 12x = 180$ Divide both sides by $12$ to find $x$ $ x = 15$ Substitute $15$ for $x$ in the expression that was given for $m\angle AOB$ $ m\angle AOB = 7({15}) - 46$ Simplify: $ {m\angle AOB = 105 - 46}$ So ${m\angle AOB = 59}$.